The numerical results are shown in Fig. REF  for the planar cases of the square lattice and the triangular tiling, which are two perfect Archimedean lattices [1]}. For the case of the square ice we have Lieb's exact result [2]} \(\epsilon ^{Exact}=(\frac{4}{3})^{3 L^2/2}\) . For the case of the triangular tiling there is no exact solution. Pauling's argument [3]} does not apply, as it relies on the bipartiteness of the lattice. For the case of the triangular tiling, we find that \(\frac{\ln \epsilon }{L^2} \ge 2.33[..]\) , using the fact that the Bethe permanent is a lower bound.
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